Why is the max a non-expansive operator?
In laymen's terms, a non-expansive operator is a function that brings points closer together or at least no further apart.

An example of a non-expansive operator is the function $f(x) = x/2$. The two numbers $0$ and $5$ are a distance of $5$ apart. The two output numbers $f(0) = 0$ and $f(5) = 2.5$ are 2.5 apart (which is smaller than $5$ apart). It is easy to see that $f$ brings everything closer together except when the two input numbers are the same: in which case, the distance between the outputs of the function at those numbers is at least no further apart than distance between the two input numbers.

$\max$  is a two-input function (or n-input, but the intuition should be clear from the 2-input case).  We can think of max as a function that maps pairs of numbers $(x, y)$ to single numbers (picking whichever of $x$ and $y$ is larger).

Suppose that we chose to measure the distance between pairs using Euclidean distance, and the distance between single numbers using the Euclidean distance as well. Here's an example:

The distance between (0,0) and (3,3) is $\sqrt{3^2 + 3^2} = \sqrt{18}$. The distance between $\max(0,0) = 0$ and $\max(3, 3) = 3$ is $\sqrt{{(\max(0,0) - \max(3, 3))}^2} = \sqrt{9} = 3$.

Let's consider the general case. The Euclidean distance between the 2D points $(a, b)$ and $(c, d)$ is $\sqrt{(a-c)^2 + (b-d)^2}$. There are four cases to consider:


Suppose that $a\geq b$ and $c \geq d$. In this case, the distance between max(a,b) and max(c,d) is just |a-c|, which is clearly at most $\sqrt{(a-c)^2 + (b-d)^2}$.
Suppose that $a\leq b$ and $c \leq d$. In this case, the distance is |b-d|, which is also at most the original distance.
Suppose that $a\geq b$ but $c \leq d$. Then the distance is |a-d|. Suppose that $a &gt; d$. Since $d &gt; c$, then $|a-d| &lt;= \sqrt{(a-c)^2 + (b-d)^2}$ since |a-d|&lt;=$\sqrt{(a-c)^2}$, and a symmetric argument holds for the case $d &gt; a$. 
$a\leq b$ but $c \geq d$, we can construct an argument identical to the one for case 3 above.


Since max is always bringing things closer together, or at least, no further apart, it is a non-expansive operator.